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Integral Calculus

Applications of Multiple Integrals

Essential guide to multiple integrals: detailed formulas for double and triple integrals in Cartesian, polar, cylindrical, and spherical coordinates. Ideal for engineering and mathematics students.

Double Integrals

Quantity General Formula In Cartesian Coordinates In Polar Coordinates
Area of a planar region S=∬RdAS = \iint_R dA ∬Rdx dy\displaystyle \iint_R dx\,dy ∬Rρ dρ dφ\displaystyle \iint_R \rho\,d\rho\,d\varphi
Surface area of z=f(x,y)z = f(x,y) S=∬R1+(∂z∂x)2+(∂z∂y)2 dAS = \iint_R \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2} \,dA ∬R1+fx2+fy2 dx dy\displaystyle \iint_R \sqrt{1 + f_x^2 + f_y^2}\,dx\,dy ∬Rρ2+ρ2fρ2+fφ2 dρ dφ\displaystyle \iint_R \sqrt{\rho^2 + \rho^2 f_\rho^2 + f_\varphi^2}\,d\rho\,d\varphi
Volume under a surface V=∬Rz dAV = \iint_R z\,dA ∬Rf(x,y) dx dy\displaystyle \iint_R f(x,y)\,dx\,dy ∬Rz(ρ,φ) ρ dρ dφ\displaystyle \iint_R z(\rho,\varphi)\,\rho\,d\rho\,d\varphi
Moment of inertia about the (x)-axis Ix=∬Ry2 dAI_x = \iint_R y^2\,dA ∬Ry2 dx dy\displaystyle \iint_R y^2\,dx\,dy ∬Rρ3sin⁡2φ dρ dφ\displaystyle \iint_R \rho^3\sin^2\varphi\,d\rho\,d\varphi
Moment of inertia about the origin (O) IO=∬R(x2+y2) dAI_O = \iint_R (x^2+y^2)\,dA ∬R(x2+y2) dx dy\displaystyle \iint_R (x^2+y^2)\,dx\,dy ∬Rρ3 dρ dφ\displaystyle \iint_R \rho^3\,d\rho\,d\varphi
Mass of a lamina with density δ(x,y)\delta(x,y) M=∬Rδ dAM = \iint_R \delta\,dA ∬Rδ(x,y) dx dy\displaystyle \iint_R \delta(x,y)\,dx\,dy ∬Rδ(ρ,φ) ρ dρ dφ\displaystyle \iint_R \delta(\rho,\varphi)\,\rho\,d\rho\,d\varphi
Center of mass coordinates (homogeneous lamina) {xc=1S∬Rx dAyc=1S∬Ry dA\begin{cases} \displaystyle x_c = \frac{1}{S}\iint_R x\,dA \\[1em] \displaystyle y_c = \frac{1}{S}\iint_R y\,dA \end{cases} {∬Rx dx dy∬Rdx dy∬Ry dx dy∬Rdx dy\begin{cases} \displaystyle \frac{\iint_R x\,dx\,dy}{\iint_R dx\,dy} \\[1em] \displaystyle \frac{\iint_R y\,dx\,dy}{\iint_R dx\,dy} \end{cases} {∬Rρ2cos⁡φ dρ dφ∬Rρ dρ dφ∬Rρ2sin⁡φ dρ dφ∬Rρ dρ dφ\begin{cases} \displaystyle \frac{\iint_R \rho^2\cos\varphi\,d\rho\,d\varphi}{\iint_R \rho\,d\rho\,d\varphi} \\[1em] \displaystyle \frac{\iint_R \rho^2\sin\varphi\,d\rho\,d\varphi}{\iint_R \rho\,d\rho\,d\varphi} \end{cases}
Notes
  • dAdA: differential area element.
  • For surface area, γ\gamma is the angle between the surface normal and the zz-axis.
  • In polar coordinates: x=ρcos⁡φx = \rho\cos\varphi, y=ρsin⁡φy = \rho\sin\varphi, dA=ρ dρ dφdA = \rho\,d\rho\,d\varphi.
  • For homogeneous laminas, the constant density δ\delta cancels out in center of mass formulas.

Triple Integrals

Quantity General Formula Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates
Volume of a solid V=∭VdVV = \iiint_V dV ∭Vdx dy dz\displaystyle \iiint_V dx\,dy\,dz ∭Vρ dρ dφ dz\displaystyle \iiint_V \rho\,d\rho\,d\varphi\,dz ∭Vr2sin⁡θ dr dθ dφ\displaystyle \iiint_V r^2\sin\theta\,dr\,d\theta\,d\varphi
Moment of inertia about the (z)-axis Iz=∭V(x2+y2) dVI_z = \iiint_V (x^2+y^2)\,dV ∭V(x2+y2) dx dy dz\displaystyle \iiint_V (x^2+y^2)\,dx\,dy\,dz ∭Vρ3 dρ dφ dz\displaystyle \iiint_V \rho^3\,d\rho\,d\varphi\,dz ∭Vr4sin⁡3θ dr dθ dφ\displaystyle \iiint_V r^4\sin^3\theta\,dr\,d\theta\,d\varphi
Mass of a solid with density δ(x,y,z)\delta(x,y,z) M=∭Vδ dVM = \iiint_V \delta\,dV ∭Vδ(x,y,z) dx dy dz\displaystyle \iiint_V \delta(x,y,z)\,dx\,dy\,dz ∭Vδ(ρ,φ,z) ρ dρ dφ dz\displaystyle \iiint_V \delta(\rho,\varphi,z)\,\rho\,d\rho\,d\varphi\,dz ∭Vδ(r,θ,φ) r2sin⁡θ dr dθ dφ\displaystyle \iiint_V \delta(r,\theta,\varphi)\,r^2\sin\theta\,dr\,d\theta\,d\varphi
Center of mass coordinates (homogeneous solid) {xc=1V∭Vx dVyc=1V∭Vy dVzc=1V∭Vz dV\begin{cases} \displaystyle x_c = \frac{1}{V}\iiint_V x\,dV \\[1em] \displaystyle y_c = \frac{1}{V}\iiint_V y\,dV \\[1em] \displaystyle z_c = \frac{1}{V}\iiint_V z\,dV \end{cases} {∭Vx dx dy dz∭Vdx dy dz∭Vy dx dy dz∭Vdx dy dz∭Vz dx dy dz∭Vdx dy dz\begin{cases} \displaystyle \frac{\iiint_V x\,dx\,dy\,dz}{\iiint_V dx\,dy\,dz} \\[1em] \displaystyle \frac{\iiint_V y\,dx\,dy\,dz}{\iiint_V dx\,dy\,dz} \\[1em] \displaystyle \frac{\iiint_V z\,dx\,dy\,dz}{\iiint_V dx\,dy\,dz} \end{cases} {∭Vρ2cos⁡φ dρ dφ dz∭Vρ dρ dφ dz∭Vρ2sin⁡φ dρ dφ dz∭Vρ dρ dφ dz∭Vzρ dρ dφ dz∭Vρ dρ dφ dz\begin{cases} \displaystyle \frac{\iiint_V \rho^2\cos\varphi\,d\rho\,d\varphi\,dz}{\iiint_V \rho\,d\rho\,d\varphi\,dz} \\[1em] \displaystyle \frac{\iiint_V \rho^2\sin\varphi\,d\rho\,d\varphi\,dz}{\iiint_V \rho\,d\rho\,d\varphi\,dz} \\[1em] \displaystyle \frac{\iiint_V z\rho\,d\rho\,d\varphi\,dz}{\iiint_V \rho\,d\rho\,d\varphi\,dz} \end{cases} {∭Vr3sin⁡2θcos⁡φ dr dθ dφ∭Vr2sin⁡θ dr dθ dφ∭Vr3sin⁡2θsin⁡φ dr dθ dφ∭Vr2sin⁡θ dr dθ dφ∭Vr3sin⁡θcos⁡θ dr dθ dφ∭Vr2sin⁡θ dr dθ dφ\begin{cases} \displaystyle \frac{\iiint_V r^3\sin^2\theta\cos\varphi\,dr\,d\theta\,d\varphi}{\iiint_V r^2\sin\theta\,dr\,d\theta\,d\varphi} \\[1em] \displaystyle \frac{\iiint_V r^3\sin^2\theta\sin\varphi\,dr\,d\theta\,d\varphi}{\iiint_V r^2\sin\theta\,dr\,d\theta\,d\varphi} \\[1em] \displaystyle \frac{\iiint_V r^3\sin\theta\cos\theta\,dr\,d\theta\,d\varphi}{\iiint_V r^2\sin\theta\,dr\,d\theta\,d\varphi} \end{cases}
Notes
  • Cylindrical coordinates: x=ρcos⁡φx = \rho\cos\varphi, y=ρsin⁡φy = \rho\sin\varphi, z=zz = z, dV=ρ dρ dφ dzdV = \rho\,d\rho\,d\varphi\,dz.
  • Spherical coordinates: x=rsin⁡θcos⁡φx = r\sin\theta\cos\varphi, y=rsin⁡θsin⁡φy = r\sin\theta\sin\varphi, z=rcos⁡θz = r\cos\theta, dV=r2sin⁡θ dr dθ dφdV = r^2\sin\theta\,dr\,d\theta\,d\varphi.
  • For homogeneous solids, the constant density δ\delta cancels out in center of mass formulas.
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